
There is a constant (hidden in the binary operator \(\lesssim \) ) such that
\(\Vert  \nabla v_h \Vert _{\Omega _h} \lesssim a_\Omega (v_h,v_h) = \Vert  v_h \Vert _{a_\Omega } \qquad v_h \in V_{h,\bigstar }\) 

where as before \(\Omega _h = \cup _{T \in \mathcal {T}_h} T\)  is the union of all the active elements, such that \(\Omega _h \supset \Omega \) .
See, respectively, (REF ) and (REF ) for these estimates in the context of aggregated and weakly stabilized methods. This property forms the basis of the effectiveness of weakly stabilized and aggregated methods. Essentially, it indicates that the solution on the extended domain \(\Omega _h\) , including the fictitious parts outside \(\Omega \)  of active elements, is controlled by \(a_\Omega (\cdot ,\cdot )\) . Consequently, this precludes the very small eigenvalues on small cut elements discussed in Section REF , and it provides control of the normal gradients which enables bounding the Nitsche parameter \(\beta \)  from above as discussed in Section REF  and REF . With aggregation this is achieved by directly precluding basis functions with a very small support in the physical domain as discussed in Section REF , while with the ghost penalty such functions are given a contribution in \(a_\Omega (\cdot ,\cdot )\)  by \(s_h(\cdot ,\cdot )\)  as discussed in Section REF .

There is an interpolation operator \(\pi _h: H^s(\Omega ) \rightarrow V_{h,\bigstar }\) , \(s\ge 2\) , such that
\(\Vert  \nabla ^m (v - \pi _h v) \Vert _{\Omega _h} \lesssim h^{s-m} \Vert  v \Vert _{H^s(\Omega )} \qquad 0\le m \le s \le p+1\) 

we use a continuous extension operator \(\mathcal {E}: v \in H^s(\Omega ) \rightarrow H^s(\mathbb {R}^d)\)  to extend \(v\)  outside of \(\Omega \) . For ghost-penalty-stabilized methods, the interpolation operator is generally constructed by first extending the function \(v\)  outside of \(\Omega \)  and then applying a standard interpolation operator, for instance the Clément operator [1]}. For the aggregated finite element space, a specific construction of the extension is used to establish the interpolation estimate [2]}, [3]}.

The stabilization form \(s_h(\cdot ,\cdot )\)  satisfies
\(\Vert  \pi _h v \Vert _{s_h}^2 = s_h(\pi _h v,\pi _h v) \lesssim h^{s-1} \Vert  v \Vert _{H^s(\Omega )} \qquad 1\le s \le p+1\) 

This is a consistency assumption on the stabilization form \(s_h(\cdot ,\cdot )\) , formulated in such a way that the stabilization form only acts directly on functions in the finite element space, and acts on other functions \(v\)  indirectly via their projection \(\pi _h{}v\) . Note that the stabilization form \(s_h(\cdot ,\cdot )\)  may for instance involve higher-order derivatives, which are not generally well defined for the exact solution or the solution to a dual problem used in an error analysis. This complication is avoided by formulating the consistency condition (REF ) in terms of projections onto the finite element space.
To provide an example we verify this consistency assumption (REF ) for the face-based
stabilization form (REF ) in the case of linear elements. Consider a face \(F \in \mathcal {F}_h\)  shared by the two elements \(T_1\)  and \(T_2\) . Then for any linear polynomial
\(w \in \mathcal {P}_1(T_1 \cup T_2)\)  we have
\(h \Vert  [ \partial _n \pi _h v ] \Vert ^2_F= h \Vert  [ \partial _n (\pi _h v - w) ] \Vert ^2_F\lesssim \sum _{i=1}^2 \Vert  \nabla (\pi _h v - w) \Vert ^2_{T_i}\) 
where we used a trace inverse estimate to pass from the face to the elements [4]}. Using the Bramble-Hilbert lemma (see, e.g., [5]}) we can choose \(w\)  in such a way that
\(\sum _{i=1}^2 \Vert  \nabla (\pi _h v - w) \Vert ^2_{T_i}\le \sum _{i=1}^2\Vert  \nabla (\pi _h v - v) \Vert ^2_{T_i} + \Vert  \nabla (v - w) \Vert ^2_{T_i}\lesssim \sum _{i=1}^2 h^2 \Vert  v \Vert ^2_{H^2(S(T_i))}\) 

where \(S(T_i)\)  is the union of the set of elements in \(\mathcal {T}_h\)  that share a node with \(T\) .

